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Quantitative Risk Management / Empirical Properties of Financial Data and GARCH Volatility Models

ARCH and GARCH Volatility Models

The stylised facts establish two properties that a good return model must have: (i) zero serial correlation in the returns themselves (the conditional mean is near zero), but (ii) persistent serial correlation in absolute returns (the conditional variance is time-varying). ARMA models capture (i) but cannot capture (ii): their variance is constant by construction. GARCH (generalised autoregressive conditionally heteroscedastic) models capture (ii) by making the conditional variance σt2=Var(Xt∣Ft−1)\sigma_t^2 = \text{Var}(X_t | \mathcal{F}_{t-1}) an autoregressive function of past squared returns and past variances.
The GARCH(1,1)(1,1) model — introduced by Bollerslev (1986) — is the single most widely used parametric volatility model in finance. It captures volatility clustering, produces a heavy-tailed unconditional distribution (even with normal innovations), and requires only three parameters. Its fitted variance sequence (σ^t2)(\hat\sigma_t^2) drives the risk-measurement pipeline: conditional Value-at-Risk (VaR) and Expected Shortfall (ES) are computed from σ^t\hat\sigma_t, and standardised residuals Xt/σ^tX_t/\hat\sigma_t are the input to filtered historical simulation.
Learning objectives. By the end of this note you should be able to:
  • write the ARCH(p)(p) and GARCH(1,1)(1,1) equations and name every parameter;
  • state the covariance stationarity conditions α1<1\alpha_1 < 1 for ARCH(1)(1) and α1+β1<1\alpha_1 + \beta_1 < 1 for GARCH(1,1)(1,1), and derive the unconditional variance;
  • explain why the squared GARCH(1,1)(1,1) process is an ARMA(1,1)(1,1) for Xt2X_t^2;
  • describe the volatility clustering mechanism in intuitive terms;
  • outline conditional maximum likelihood estimation and quasi-MLE;
  • state what variance targeting is and when it is used.

In this lesson

    1. ARCH processes
    1. GARCH(1,1)(1,1)
    1. Maximum likelihood estimation
    1. Reading a GARCH output

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